Flash Shit :3
17/09/2017 at 08:38-
Call the formular of numbers we need to find is \(\overline{abcd}\)
Condition : m + k < 200
Following the thread, we have :
\(\left\{{}\begin{matrix}\overline{abcd}=k^2\\\overline{\left(a+1\right)\left(b+3\right)\left(c+5\right)\left(d+3\right)}=m^2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\overline{abcd}=k^2\\\overline{abcd}+1353=m^2\end{matrix}\right.\)
=> m2 - k2 = 1353
<=> (m - k)(m + k) = 1353 = 123.11 = 41.33
=> \(\left\{{}\begin{matrix}m+k=123\\m-k=11\end{matrix}\right.\) or \(\left\{{}\begin{matrix}m+k=41\\m-k=33\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}m=67\\k=56\end{matrix}\right.\) or \(\left\{{}\begin{matrix}m=37\\k=4\end{matrix}\right.\)
With m = 67 ; k = 56
=> \(\left\{{}\begin{matrix}\overline{abcd}=k^2=3136\\\overline{abcd}=m^2-1353=3136\end{matrix}\right.\Rightarrow\overline{abcd}=3136\)
With m = 37 , k = 4
=> .......... (This case is disqualified , you can solve that equation)
So , the number we need to find is 3136
Selected by MathYouLike -
Faded 28/01/2018 at 20:43Call the formular of numbers we need to find is ¯¯¯¯¯¯¯¯¯¯abcd
Condition : m + k < 200
Following the thread, we have :
{¯¯¯¯¯¯¯¯¯¯abcd=k2¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(a+1)(b+3)(c+5)(d+3)=m2
⇔{¯¯¯¯¯¯¯¯¯¯abcd=k2¯¯¯¯¯¯¯¯¯¯abcd+1353=m2
=> m2 - k2 = 1353
<=> (m - k)(m + k) = 1353 = 123.11 = 41.33
=> {m+k=123m−k=11
or {m+k=41m−k=33
=> {m=67k=56
or {m=37k=4
With m = 67 ; k = 56
=> {¯¯¯¯¯¯¯¯¯¯abcd=k2=3136¯¯¯¯¯¯¯¯¯¯abcd=m2−1353=3136⇒¯¯¯¯¯¯¯¯¯¯abcd=3136
With m = 37 , k = 4
=> .......... (This case is disqualified , you can solve that equation)
So , the number we need to find is 3136